UROP Proceedings 2020-21

School of Science Department of Mathematics 45 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: WU Ruirui / MATH Course: UROP2100, Fall This report will briefly present theorems proved in the paper by Hamilton in 1986 and paper by Brendle and Schoen in 2008. These two papers studied manifolds with specific curvature conditions and proved them to be diffeomorphic to some sphere space forms. The report will mainly focus on how the proof of main theorems goes in them, and complete some details omitted by authors. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: CHEN Zijun / MATH-PMA LIU Lingchong / MATH-PMA Course: UROP3100, Fall UROP3100, Fall In this semester, we went through RICHARD S. HAMILTON’s paper FOURMANIFOLDS WITH POSITIVE CURVATURE OPERATOR. And then discuss definition and application of PIC(Pinched Isotropic Curvature) in MANIFOLDS WITH 1/4-PINCHED CURVATURE ARE SPACE FORMS, by SIMON BRENDLE and RICHARD SCHOEN. We spent most of our time discussing and understanding the ODE-PDE Maximum principle and convergence criterion mentioned in Hamilton’s paper, and then applied it on the dimension 3 and dimension 4 cases. In this report, we will try to summarize above discussed contents from the perspective of logic relations between positivity of different curvatures and then. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: LI Jiangtao / MATH-PMA Course: UROP3100, Spring In Hamilton’s 1982 paper, he introduced the Ricci Flow equation = −2Ric(g) and proved that the psoitive Ricci curvature is preserved by Ricci Fow, with help of a special maximum principle (the maximum principle for tensor). In this expository paper, we study the effect of Ricci flow on another important curvature condition, nonnegative isotropic curvature and follow Nguyen to prove that it is also preserved by Ricci Flow. However, this curvature condition is not as simple as positive Ricci curvature. To handle with this, we need to introduce a modified version of maximum principle, which allows us to reduce the study of PDE on vector bundle over manifold to system of ODEs on fibers of the bundle. This maximum principle was also first introduced by Hamilton in his 1986 paper and then Nguyen and Brendle, with Schoen used it to prove that nonnegative isotropic curvature is preserved by Ricci Flow. Brendle and Schoen went further and finally proved the differentiable sphere theorem, which we will discuss briefly in this paper but will not present the proof due to limited research time. At the end of the paper, by a modified version differentiable sphere theorem due to Petersen and Tao, we deduced a improved differentiable sphere theorem for odd dimension in Corollary5.4.